The given book presents an introduction to the basic concepts and mathematical tools of quantum mechanics. It is based on the material that I have been using in teaching the first course on quantum mechanics to the undergraduate and M.Sc. students at I. I. T. Delhi. The last chapter on relativistic generalization of quantum mechanics does not constitute a part of the usual course and has been added for those who wish to have some basic ideas of relativistic quantum mechanics.

In presenting the material, I have taken into account the feedback of the students about the conceptual as well as the mathematical difficulties faced by them during the course. As a result, I have tried to be as simple as possible. Therefore, I might appear to be too simple and repetitive at times and I hope the knowledgeable reader will pardon me for that.

The book starts with the basics of quantum mechanics in the traditional way by using the fundamental tools of mathematical analysis with an emphasis on the physical explanation for the mathematical treatment of the topics. This part includes the introduction to the concept of the state of a quantum mechanical system, operators and their algebra, the basic postulates of quantum mechanics and the solution of the Schrödinger equation for important one-dimensional systems. The algebraic formalism in the traditional language of Dirac is then introduced and the entire earlier material is reformulated in this language so as to make the reader comfortable with the modern language of quantum mechanics. In the later chapters of the book, I deal with the three-dimensional problems, hydrogen atom, quantum mechanical theory of orbital as well as spin angular momentum, and many particle systems. Simple effects related to the quantum mechanical treatment of the motion of a charged particle in the presence of a magnetic field are also presented. The basic concepts related to the symmetries of a system and the corresponding laws of conservation are then introduced and developed. In particular, the relationship between the fundamental quantum mechanical operators and the generators of the continuous groups of symmetries of spacetime are established and discussed. The book ends with an introduction to relativistic quantum mechanics.

According to Herman Wey1, by symmetry of an object (or a physical system) we mean the property of the object to appear unchanged after some operation has been done on it. We then say that the object is symmetrical under the given operation. For instance, consider a square. It is indistinguishable after rotations by and about the axis passing through its geometrical center and perpendicular to its plane (Shown by the dot in the figure). This axis is said to be the axis of symmetry of the square. Note that the angle of rotation, for which the square possesses symmetry, takes on only discrete values. Consequently, it has, as we say, a discrete symmetry. On the other hand, a sphere looks unchanged after all rotations (infinitesimal or finite) about its axis of symmetry. Since the angle of rotation can take continuous values, the rotational symmetry of the sphere is a continuous symmetry.

It turns out that, for each continuous symmetry of a physical system, there exists a conserved quantity, i.e., a physical characteristic that remains constant as the system evolves in time according to a given dynamical equation. This result is known as the celebrated Nöther theorem. For example, if we place a system of particles in empty space, far from anything that might affect it, it does not make a difference where exactly we put it. There are no preferred locations in empty space; all locations are equivalent. As a consequence, there is a symmetry for a system of particles with respect to translations in empty space. This translational symmetry leads to the law of conservation of the total linear momentum of the system. Similarly, there exists a symmetry for a system of particles in empty space with respect to rotations of the system as a whole because there are no preferred directions in empty space. This rotational symmetry leads to conservation of the total angular momentum of the system. Another important symmetry is the symmetry with respect to shift in time. It turns out that it does not matter when we perform an experiment on an isolated system. The results will be the same. This symmetry with respect to shift in the origin of time gives rise to the law of conservation of energy.